Three-dimensional Rayleigh-Taylor instability (RTI) with the time-varying acceleration in a finite domain is investigated in a systematic framework. The acceleration magnitude follows a power law in time with an exponent greater than −2. Applying the group theory, the instabilities are demonstrated considering the irreducible representations for observable periodic structures with a square symmetry in the plane normal to the acceleration. We derive the dynamical system and illustrate the universal form of the solutions in the linear and nonlinear regimes. The scale-dependent dynamics are shown to be single scale and multiscale in the two regimes, respectively. For the nonlinear regime solutions, fundamental scales are derived bridging the solutions in the finite- and infinite-sized domains. Special solutions for bubbles and spikes are identified from a one-parameter family of solutions. The effect of domain confinement is that the velocity and curvature decreases and shear increases as the domain size is reduced. The theory provides predictions for the flow field and demonstrates the interfacial behavior of RTI. Our results are in good agreement with the prior studies and also provide new benchmarks for experiments and simulations.